Application of the covering space in the complex integral of multiply connected domains
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摘要: 众所周知，单连通区域上解析函数所确定的变上限积分是一个单值函数，然而对于多连通区域D上解析函数f（z）的变上限积分$F\left( z \right) = \int_{{z_0}}^z {f\left( \zeta \right)} {\rm{d}}\zeta $，F（z）不仅依赖于z（z0是D内固定的一点），还依赖以下两点：（1）积分的路径；（2）函数f（z）关于洞是否恰当.由此可以知道F（z）可能是一个多值函数.以上结果均可以在一般复变函数教材中找到，这里不再赘述.本文利用黎曼曲面的正则覆盖曲面知识，给出了解析函数f（z）在多连通区域上积分的一种新诠释.Abstract: It is well known that the integral with variable upper limit of analytic function is a single value function in the simple connected domain, while the integral with variable upper limit of analytic function in the multiply connected domains is as following: $F\left( z \right) = \int_{{z_0}}^z {f\left( \zeta \right)} {\rm{d}}\zeta $, F(z) is not only dependent on the z (z0 is the fixed point in D), but also depends on the integral path and function f(z) being exact or not in every hole. Therefore F(z) is likely to be multiple valued function. In this paper, we give a new proof method about the integral of analytic function f(z) in the multiply connected domain by the regular covering surface.
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[1] 吕以辇, 张学莲.黎曼曲面[M].北京:科学出版社, 1997. [2] 聂灵沼, 丁石孙.代数学引论[M].北京:高等教育出版社, 2010. [3] 熊金城.点集拓扑讲义[M].北京:高等教育出版社, 2003. [4] 余家荣.复变函数[M].北京:高等教育出版社, 2007. -
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